The Princeton Companion to Mathematics mentions that polynomials (for instance, ones with rational coefficients) share similarities with integers, thus leading to the idea of a general structure of the Euclidean domain. It isn't obvious to me how this is the case. Could you provide a palatable explanation?

Euclidean domains are rings that can be endowed with the structure of an Euclidean function. An example is the ring of polynomials $\mathbb{K}(X)$ over a field $\mathbb{K}$, endowed with the Euclidean function $f(P)=\mathrm{deg}P$, $P\in\mathbb{K}(X)$.
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user122283Apr 24 '14 at 1:49